Correlation Entropy and Power-Law Kinetics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Author of Manuscript entropy-4389569, titled: Correlation Entropy and Power-Law Kinetics.

I recommend to accept the manuscript  entropy-4389569 after a minor revision, because:

• This is a research work where for the first time the term Correlation Entropy is introduced in Chemical-Physical Kinetics resulting in a new approach to study & apply the Arrhenius & other kinetic frameworks unified in a single thermodynamic equation.
• Therefore, the author identify three well-known different kinetic mechanisms: the cooperative evolution, the self-limiting evolution, & the independent evolution arising from a common thermodynamic equation. Thus, the collision theory that goes in the activation energy is not used in the manuscript & only a thermodynamic unified description is constructed.
• Mathematically, these three situations become limiting cases of χ (chi), an slope which enters into Equation (28). Thus, d((ΔG - ΔH)/ln N) = - χ  where N is a cumulative number of microstates in the correlation entropy (equation 6) with ΔS_corr =  κ χ Ln N
• Finally, this is an important research work. Despite the author does not mention it, this somehow is related to the very difficult stiffness problem for the characteristic times of a reaction which can vary many orders of magnitude, because the dependent variables change very fast & the numerical solution is very slow.

For the Author, I attach the PDF where in orange are highlighted some suggestions in the PDF pop-ups, where I give the questions.

Otherwise, the blue, green, & violet highlights are to follow the manuscript by myself.

1. Please, is A constant in equation (1)? Line 168.
2. Please, the 3 colors (red,blue & green could be used in the caption of figure 1. Lines 235 - 239.
3. Please, it could be useful to name the colors in Fig. 1 which represent each kind of kinetic evolution behavior. Lines 235 - 239. 
4. Please, Do you mean the value of the kinetic exponents? Lines 248 - 249.
5. Please, could you briefly explain the words: fatigue, trapping, & degradation. From the paragraph in lines 253- 255 is not clear.
6. Please, could it be useful here to include the Coffin-Manson relationship in the text. lines 265 - 267.
7. Please, is it there an specific Equation for this statement, and its equation? in lines 269 - 272.

8. Please, does this case happen for systems with an physical effective self-consistent effective field, such as. the case of the stiffness problem in Chemical Kinetics? Lines 278 - 284. Check for example: John W. Daily. Statistical Thermodynamics: An Engineering approach, Cambridge University Press, 2019. starting from page 230.

9. This phrase by itself needs a reference because is important to the topic of your manuscript. Lines 292 - 293.
10. Please, is this case mathematically described numerically by a fixed point self-consistency which is not so difficult to solve? Lines 304 - 306. (I apologise if this question is incorrect).
11.  Please, ¿Bias temperature instability = BTI? Line 313
12. Please, screening effects occurs when there is a strong dependence of previous values obtained in the coefficients, and represent an effective field? Line 322
13. Important statement, some references could be added here. Lines 326 - 328
14. Please, could you add a reference related to your case here. This is important, because disorder and its interaction occurs in the kinetics of several superconducting alloys, and it is very difficult to obtain and describe properly. Line 329
15. The free Gibbs energy sometimes contains an external field that could be an effective self-consistent field, where the kinetic description becomes quite complicate. lines 336 - 338. 
16. Please, is this the definition that introduces the correlation constant (𝜒) ?lines 344 - 347.
17. Please, how do you define the degree of correlation in your work? I guess that is important to clarify this new quantity. Line 441.
18. α has energy units, I guess is important to define it. Equation (31) in line 616.

The referee.

Comments for author File: Comments.pdf

Author Response

 

We thank the reviewer for the careful reading of the manuscript, the positive assessment of the work, and the many constructive suggestions.

Comment 1:
Please, is A constant in equation (1)?

Response:
The manuscript has been revised to clarify that A is generally a stress- and temperature-dependent prefactor and is not necessarily a universal constant.

Comment 2 and 3:
Please identify the colors in Figure 1 and use them in the caption.

Response:
The Figure 1 caption has been revised to explicitly identify the red, blue, and green curves and associate them with accelerating, independent, and self-limiting kinetic evolution.

Comment 4:
Please, do you mean the value of the kinetic exponents?

Response:
The wording has been revised to explicitly refer to the values of the kinetic exponents.

Comment 5:
Please briefly explain fatigue, trapping, and degradation.

Response:
Additional explanatory text has been added to clarify these examples and their relationship to cooperative and self-limiting kinetic evolution.

Comment 6:
Could it be useful to include the Coffin-Manson relationship?

Response:
A discussion of Coffin-Manson fatigue behavior has been added as an example of accelerating evolution and positive correlation.

Comment 7:
Is there a specific equation for this statement?

Response:
Additional equation references have been included where appropriate to improve traceability between the discussion and the mathematical development.

Comment 8:
Does this occur in systems with a self-consistent effective field?

Response:
The reviewer raises an interesting possibility. Additional discussion has been added noting that the framework may be applicable to systems exhibiting feedback, screening, interaction, or self-consistent field effects, although a detailed treatment remains beyond the scope of the present work.

Comment 9:
This phrase needs a reference.

Response:
Additional references have been added.

Comment 10:
Could this be described by a fixed-point self-consistency approach?

Response:
A brief discussion has been added acknowledging that self-consistent formulations may provide one possible microscopic realization of the proposed correlation mechanism.

Comment 11:
Bias Temperature Instability = BTI?

Response:
BTI is now defined at first use.

Comment 12:
Screening effects and effective fields?

Response:
Additional clarification has been added regarding screening, occupancy, relaxation, and related correlation mechanisms.

Comment 13:
Important statement; references could be added.

Response:
Additional supporting references have been included.

Comment 14:
Could you add a reference related to disorder?

Response:
Additional references have been added discussing disorder-related kinetic evolution.

Comment 15:
External fields and Gibbs free energy.

Response:
The revised manuscript now explicitly includes a stress-dependent entropy contribution that contributes to the Gibbs free energy and provides a thermodynamic interpretation of stress acceleration.

Comment 16:
Is this the definition that introduces χ?

Response:
The definition of the Correlation Constant χ has been clarified and highlighted in the revised manuscript.

Comment 17:
How do you define the degree of correlation?

Response:
The manuscript now explains that χ is a phenomenological measure of the extent to which prior microstate evolution influences the accessibility of future microstates. Positive χ corresponds to cooperative evolution, negative χ corresponds to self-limiting evolution, and χ = 0 corresponds to statistically independent evolution.

Comment 18:
α has energy units and should be defined.

Response:
The parameter α is now explicitly defined as an energy scale associated with the experimentally observed temperature dependence of the kinetic exponent.

We thank the reviewer again for the many helpful suggestions.

Reviewer 2 Report

Comments and Suggestions for Authors

Referee report Entropy 4389569

J. B. Bernstein

Correlation Entropy and Power-Law Kinetics

   The author proposes a generalized framework in which power-law behaviour emerges naturally by introducing a correlation entropy term based on statistical mechanics. A dimensionless correlation constant, χ, is introduced to quantify the influence of accumulated microstate evolution. We propose an extension of the statistical definition of entropy that includes a correlation entropy term, 𝛥𝑆_corr = kχ ln(N), where N represents the accumulated number of microstates participating in the evolution process. This leads to a corresponding correlation-energy contribution that modifies the Gibbs free energy. The proposed framework provides a thermodynamic interpretation of the power-law exponent, establishes a direct connection between entropy, free energy, and kinetic evolution, and offers a unified description applicable to degradation, relaxation, diffusion, fatigue, and other evolving processes.

   The paper is written in a pedagogic way with a slow and non-mathematical progression introducing the concepts and ideas successively. On p.8 the author writes: “The existence of all three regimes; accelerating, independent, and self-limiting evolution raises a fundamental question. Are these three kinetic regimes fundamentally different phenomena requiring separate descriptions, or do they represent different manifestations of a common underlying thermodynamic framework? The remainder of this work explores this hypothesis.”

   The historical development of the author’s theme is reviewed in section 1, and an experimental motivation for his further development and point of view is given in section 2. This functions well for the reader. The investigation of the author begins in section 3 with the title “Thermodynamic Origin of Correlation Entropy”.

   Here is the main new point in the article: “The experimental observations discussed in Section 2 suggest that the probability of future evolution may depend upon the current state of the system. To describe this behaviour, we define here a dimensionless Correlation Constant, 𝜒. This constant characterizes the degree to which prior microstate accumulation influences subsequent evolution. Positive values correspond to cooperative behaviour (positive correlation), negative values correspond to self-limiting behaviour (negative correlation), and 𝜒 = 0 corresponds to evolution that is independent of the current state.

   The language is good, and I find no printing errors.

   My conclusion is that this paper presents a very fine work in a good way. In my opinion the paper is suitable for publication in the journal Entropy without any changes.

Author Response

Response to Reviewer 2

We sincerely thank the reviewer for the careful reading of the manuscript and for the very positive assessment.

We appreciate the reviewer's observation that the manuscript develops the ideas gradually and pedagogically and that the historical and experimental motivation provide an effective foundation for the theoretical development.

We are especially grateful for the reviewer's conclusion that:

"The paper is suitable for publication in the journal Entropy without any changes."

Although no revisions were requested, the manuscript has nevertheless been improved in response to comments from the other reviewers, including clarification of the theoretical foundations, reduction of self-citations, addition of comparisons with existing entropy frameworks, and improved discussion of experimental validation.

We thank the reviewer for the encouraging evaluation of the work.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have developed a new theoretical framework and aim to explain the widespread occurrence of power-law kinetics through a novel thermodynamic mechanism. They propose that power-law kinetics observed in different fields share a common thermodynamic origin. The following major revisions should be considered:

1. How does the proposed framework differ from existing approaches such as Tsallis entropy, Rényi entropy, non-extensive thermodynamics, or other generalized statistical-mechanics frameworks?

2. How does the proposed theory compare with these established explanations, and what specific gap in the literature does it fill?

3. What is the fundamental theoretical basis for Eq. (6)? Is it derived from statistical mechanics, or is it introduced as a phenomenological postulate?

4. Which specific shortcomings of existing theories motivate the introduction of the present framework?

5. How can the Correlation Constant (χ) be measured independently? Please elaborate.

6. Under what conditions could the theory be falsified? Please elaborate it.

7. What observations would contradict the proposed framework? Please elaborate it.

8. Can the author identify at least one experimentally observed phenomenon that is explained uniquely or more effectively by the proposed framework?

Comments on the Quality of English Language

 The English could be improved to more clearly express the research.

Author Response

Response to Reviewer 3

We thank the reviewer for the thoughtful and constructive comments. The questions raised significantly improved the manuscript by helping us clarify the scope, motivation, and interpretation of the proposed framework.

Comment 1:
How does the proposed framework differ from Tsallis entropy, Rényi entropy, non-extensive thermodynamics, and related approaches?

Response:
A new discussion has been added to Section 3 introducing both Tsallis entropy and Rényi entropy. We clarify that these approaches modify the mathematical form of the entropy measure itself. In contrast, the present work retains the classical Boltzmann relationship

S = k ln(Ω)

and introduces a phenomenological correlation-dependent contribution through the accessibility of microstates. Thus, the manuscript does not propose a new entropy formalism but rather a new thermodynamic interpretation of correlated kinetic evolution.

Comment 2:
How does the theory compare with established explanations and what gap does it fill?

Response:
The revised manuscript now explicitly discusses this issue. Existing approaches successfully describe specific classes of systems but generally do not provide a unified thermodynamic interpretation of accelerating, independent, and self-limiting power-law kinetics. The present framework attempts to provide a common thermodynamic description applicable across multiple fields.

Comment 3:
What is the theoretical basis of Eq. (6)?

Response:
Section 3 has been substantially revised. A new discussion introduces the decomposition

Ω = Ω_Ψ Ω_corr

where ΩΨ represents stress-accessible microstates and Ωcorr represents additional accessibility arising from correlation effects. We emphasize that Eq. (6) is introduced as a phenomenological hypothesis motivated by correlated kinetic evolution rather than as a rigorous derivation from first principles. This clarification is now explicitly stated in the manuscript.

Comment 4:
What shortcomings of existing theories motivate the framework?

Response:
The revised manuscript now emphasizes that many observed power-law behaviors are treated empirically through fitting exponents without a corresponding thermodynamic interpretation. The present work seeks to provide a physical interpretation of those exponents through Gibbs free energy and entropy.

Comment 5:
How can χ be measured independently?

Response:
Additional discussion has been added showing that χ is related to the experimentally measurable power-law exponent through

χ = 1 − m

where

m = 1/n.

Thus χ can be extracted directly from observed kinetic evolution. The manuscript further discusses how χ may ultimately be connected to independently measurable microscopic correlation mechanisms.

Comment 6:
Under what conditions could the theory be falsified?

Response:
A new discussion has been added describing falsifiability. The framework would be challenged if systems exhibiting identical values of χ consistently exhibited fundamentally different kinetic behavior, or if independently measured correlation mechanisms were found to be unrelated to the extracted χ values.

Comment 7:
What observations would contradict the framework?

Response:
Additional discussion has been added indicating that observations demonstrating no relationship between extracted χ values and the observed evolution would weaken the proposed interpretation. Such observations would require revision or replacement of the hypothesis.

Comment 8:
Can the author identify a phenomenon explained uniquely or more effectively by the framework?

Response:
The revised manuscript now discusses Bias Temperature Instability (BTI) as an example in which experimentally observed temperature dependence of the power-law exponent can be interpreted through the correlation-energy framework. This leads to the observation that the inferred correlation-energy scale may remain approximately temperature independent even when the exponent itself varies strongly with temperature.

Comment on English Language

Response:
The manuscript has been extensively edited for clarity, terminology, consistency, and readability. Several sections were rewritten, notation was standardized, and additional explanatory text was added throughout.

We thank the reviewer for helping improve the presentation and clarity of the manuscript.